Means to make every possible effort. The popular belief is that this phrase derives from the manner of construction of pipe organs. These instruments have stops to control the air flow through the pipes and pulling them out increases the musical volume.
Prior to the introduction of pipe organs the word 'stop' had, in a musical context, been used to mean 'note' or 'key'. That usage is recorded as early as the late 16th century. The word 'stop' later came to be used for the knobs that control the flow of air in pipe organs, by pushing them in or pulling them out.